The history of statistics is filled with many controversies, in which the prime focus has been the difference in the “interpretation of probability” between Fre- quentist and Bayesian theories. Many philosophical arguments have been elabo- rated to examine the problems of both theories based on this dichotomized view of statistics, including the well-known stopping-rule problem and the catch-all hy- pothesis problem. However, there are also several “hybrid” approaches in theory, practice, and philosophical analysis. This poses many fundamental questions. This paper (...) reviews three cases and argues that the interpretation problem of probabil- ity is insufficient to begin a philosophical analysis of the current issues in the field of statistics. A novel viewpoint is proposed to examine the relationship between the stopping-rule problem and the catch-all hypothesis problem. (shrink)

Our paper studies the anatomy of the discovery of the Higgs boson at the Large Hadron Collider and its influence on the broader model landscape of particle physics. We investigate the phases of this discovery, which led to a crucial reconfiguration of the model landscape of elementary particle physics and eventually to a confirmation of the standard model. A keyword search of preprints covering the electroweak symmetry breaking sector of particle physics, along with an examination of physicists own understanding of (...) the discovery as documented in semiannual conferences, has allowed us an empirical investigation of its model dynamics. From our analyses we draw two main philosophical lessons concerning the nature of scientific reasoning in a complex experimental and theoretical environment. For one, from a confirmation standpoint, some SM alternatives could be considered even more confirmed by the Higgs discovery than the SM. Nevertheless, the SM largely remains the commonly accepted account of EWSB. We present criteria for comparing degrees of confirmation and expose some limits of a purely logical approach to understanding the Higgs discovery as a victory for the SM. Second, we understand the persistence of SM alternatives in the face of disfavourable evidence by borrowing the Lakatosian concept of a research programme, where the core idea behind a group of models survives, while other aspects adapt to incoming data. In order to apply this framework to the model landscape of EWSB, we must introduce a new category of research programme, the model-group, and we test its viability using the example of composite Higgs models. (shrink)

If there are two dependent positive real variables x1 and x2, and only x1 is known, what is the probability that x2 is larger versus smaller than x1? There is no uniquely correct answer according to “frequentist” and “subjective Bayesian” definitions of probability. Here we derive the answer given the “objective Bayesian” definition developed by Jeffreys, Cox, and Jaynes. We declare the standard distance metric in one dimension, d(A,B)≡|A−B|, and the uniform prior distribution, as axioms. If neither variable is known, (...) P(x2x1)=P(x2>x1). This appears obvious, since the state spaces x2x1 and x2>x1 have equal size. However, if x1 is known and x2 unknown, there are infinitely more numbers in the space x2>x1 than x2x1. Despite this asymmetry, we prove P(x2x1∣x1)=P(x2>x1∣x1), so that x1 is the median of p(x2|x1), and x1 is statistically independent of ratio x2/x1. We present three proofs that apply to all members of a set of distributions. Each member is distinguished by the form of dependence between variables implicit within a statistical model (gamma, Gaussian, etc.), but all exhibit two symmetries in the joint distribution p(x1,x2) that are required in the absence of prior information: exchangeability of variables, and non-informative priors over the marginal distributions p(x1) and p(x2). We relate our conclusion to physical models of prediction and intelligence, where the known ’sample’ could be the present internal energy within a sensor, and the unknown the energy in its external sensory cause or future motor effect. (shrink)